A fractional calculus concept is considered in the framework of a Volterra type integro-differential equation, which is employed for the self-consistent description of the high-gain free-electron laser (FEL). It is shown that the Fox H-function is the Laplace image of the kernel of the integro-differential equation, which is also known as a fractional FEL equation with Caputo–Fabrizio type fractional derivative. Asymptotic solutions of the equation are analyzed as well.
The optimal parameter space for an X-ray free-electron laser (FEL) in the self-amplified spontaneous emission (SASE) operation mode is examined. This study focuses on FEL operation with a shorter undulator period and higher undulator strength made available through recent developments in in-vacuum, cryogenic and superconducting undulators. Progress on short-period undulator technologies is surveyed and FEL output characteristics versus undulator parameters are computed. The study is performed on a case of the planned soft-X-ray FEL at the MAX IV Laboratory in Sweden. An extension of the SASE mode into the harmonic lasing self-seeded mode is also analysed.
Fox H-Functions in Self-Consistent Description of a Free-Electron Laser